On attainability of optimal controls in coefficients for system of Hammerstein type with anisotropic p-Laplacia

0.37 MB·

by  T. Durante

Ricerche di Matematica manuscript No.
(will be inserted by the editor)
On attainability of optimal controls in coefficients
for system of Hammerstein type with anisotropic
p-Laplacian
Tiziana Durante · Olha P. Kupenko ·
Rosanna Manzo
7
1
0
Received: date/Accepted: date
2

n
a
Abstract In this paper we consider an optimal control problem (OCP) for
J
thecoupledsystemofanonlinearmonotoneDirichletproblemwithanisotropic
3
p-Laplacianandmatrix-valuedL∞(Ω,RN×N)-controlsinitscoefficientsanda
2
nonlinear equation of Hammerstein type. Using the direct method in calculus

] ofvariations,we provethe existence of anoptimalcontrolin consideredprob-
C
lem and provide sensitivity analysis for a specific case of considered problem
O
with respect to two-parameter regularization.
.
h
t Keywords Nonlinear elliptic equations Hammerstein equation control in
a
· ·
m coefficients p(x)-Laplacian approximation approach
· ·
[

Mathematics Subject Classification (2000) 47H30 35B20 35M12
1 · · ·
35J60 49J20
v
·
9
0
T.Durante
6
Universit`adegliStudidiSalerno,DipartimentodiIngegneriadell’Informazione
6
edElettricaeMatematicaApplicata,
0
ViaGiovanni PaoloII,132,84084Fisciano(SA),Italy
.
1 E-mail:[email protected]
0
O.Kupenko
7 NationalMiningUniversity,DepartmentofSystemAnalysisandControl,
1 Yavornitskyiav.,19,49005Dnipro,Ukraine,
: NationalTechnical UniversityofUkraine“KievPolytechnical Institute”,
v
InstituteforAppliedandSystem Analysis,
i
X Peremogyav.,37,build.35,03056Kiev,Ukraine
E-mail:kogut [email protected]
r
a R.Manzo
Universit`adegliStudidiSalerno,DipartimentodiIngegneriadell’Informazione
edElettricaeMatematicaApplicata,
ViaGiovanni PaoloII,132,84084Fisciano(SA),Italy
E-mail:[email protected]
2 T.Durante,O.P.Kupenko,R.Manzo
1 Introduction
The aim of this paper is to prove the existence result for an optimal control
problem (OCP) governed by the system of a homogeneous Dirichlet nonlin-
ear elliptic boundary value problem, whose principle part is an anisotropic
p-Laplace-like operator, and a nonlinear equation of Hammerstein type, and
to provide sensitivity analysis for the specific case of considered optimiza-
tion problem with respect to a two-parameter regularization. As controls we
consider the symmetric matrix of anisotropy in the main part of the elliptic
equation. We assume that admissible controls are measurable and uniformly
bounded matrices of L∞(Ω;RN×N).
Systemswithdistributedparametersandoptimalcontrolproblemsforsys-
tems describedbyPDE,nonlinearintegralandordinarydifferentialequations
havebeenwidelystudiedbymanyauthors(seeforexample[13,19,21,23,27]).
However, systems which contain equations of different types and optimiza-
tion problems associated with them are still less well understood. In general
case including as well control and state constraints, such problems are rather
complex and have no simple constructive solutions. The system, considered
in the present paper, contains two equations: a nonlinear elliptic equation
with the so-calledanisotropicp-Laplace operator with homogeneousDirichlet
boundary conditions and a nonlinear equation of Hammerstein type, which
nonlinearly depends on the solution of the first object. The optimal control
problem we study here is to minimize the discrepancy between a given dis-
tribution z Lp(Ω) and a solution of Hammerstein equation z = z(A,y),
d
choosing an a∈ppropriate matrix of coefficients A A , i.e.
ad
∈
I(A,y,z)= z(x) z (x)2dx inf (1)
d
| − | −→
ZΩ
subject to constrains
z+BF(y,z)=0 in Ω, (2)
div (A(x) y, y)RN (p−2)/2A(x) y =f in Ω, (3)
− | ∇ ∇ | ∇
A A , y =0 on ∂Ω, (4)
(cid:0) ad (cid:1)
∈
where B : Lq(Ω) Lp(Ω) is a positive linear operator, F : W1,p(Ω)
→ 0 ×
Lp(Ω) Lq(Ω) is a nonlinear operator, f L2(Ω) is a given distribu-
tion, an→d a class of admissible controls A is∈a nonempty compact subset
ad
of L∞(Ω;RN(N2+1)).
The interestto equationswhose principle partis ananisotropicp-Laplace-
likeoperatorarisesfromvariousappliedcontextsrelatedtocompositemateri-
als such as nonlinear dielectric composites, whose nonlinear behavior is mod-
eledbytheso-calledpower-low(see,forinstance,[4,20]andreferencestherein).
Itissufficienttosaythatanisotropicp-Laplacian∆ (A,y)hasprofoundback-
p
ground both in the theory of anisotropic and nonhomogeneous media and in
Finsler or Minkowski geometry [31]. As a rule, the effect of anisotropy ap-
pearsnaturallyinawideclassofgeometry—Finslergeometry.Atypicaland
OptimalcontrolsforHammersteinsystemwithanisotropicp-Laplacian 3
important example of Finsler geometry is Minkowski geometry. In this case,
anisotropicLaplacianis closelyrelatedtoaconvexhypersurfaceinRN,which
iscalledtheWulffshape[30].SincethetopologyoftheWulffshapeessentially
depends on the matrix of anisotropy A(x), it is reasonable to take such ma-
trix as a control.Frommathematicalpointof view,the interestof anisotropic
p-Laplacian lies on its nonlinearity and an effect of degeneracy, which turns
out to be the major difference from the standard Laplacian on RN.
Inpractice,theequationsofHammersteintypeappearasintegralorinteg-
ro-differential equations. The class of integral equations is very important for
theory and applications, since there are less restrictions on smoothness of the
desired solutions involved in comparison to those for the solutions of differ-
ential equations. It should be also mentioned here, that well posedness or
uniqueness of the solutions is not typical for equations of Hammerstein type
or optimization problems associated with such objects (see [3]). Indeed, this
property requires rather strong assumptions on operators B and F, which is
ratherrestrictiveinviewofnumerousapplications(see[25]).Thephysicalmo-
tivationofoptimalcontrolproblemswhicharesimilarto thoseinvestigatedin
the present paper is widely discussed in [3,26].
Using the direct method of the Calculus of Variations, we show in Section
4 that the optimal control problem (1)–(4) has a nonempty set of solutions
provided the admissible controls A(x) are uniformly bounded in BV-norm,
in spite of the fact that the corresponding quasilinear differential operator
div (A y, y)RN p−22A y ,inprinciple,hasdegeneraciesas A12 y tends
− | ∇ ∇ | ∇ | ∇ |
p−2
to ze(cid:0)ro [1]. Moreover, when(cid:1)the term |(A∇y,∇y)RN| 2 is regarded as the
coefficientofthe Laplaceoperator,wehavethe caseofunbounded coefficients
(see [12,14]). In order to avoid degeneracy with respect to the control A(x),
we assume that matrix A(x) has a uniformly bounded spectrum away from
zero.As for the optimalcontrolproblems incoefficientsfor degenerateelliptic
equations and variational inequalities, we can refer to [5,8,9,10,16,17,19].
A number of regularizations have been suggested in the literature. See
[24] for a discussion for what has come to be known as (ε,p)-Laplace prob-
lem, such as div((ε+ y 2)p−22) y. While the (ε,p)-Laplacian regularizes
− |∇ | ∇
the degeneracy as the gradients tend to zero, the term y p−2, viewed again
|∇ |
as a coefficient, may grow large [6]. Therefore, following ideas of [7], for the
specificcaseofconsideredoptimizationproblemweintroduceyetanotherreg-
ularizationthat leads to a sequence of monotone and bounded approximation
k(A21 y 2)of A12 y 2. As a result,forfixed parameterp [2, )andcon-
F | ∇ | | ∇ | ∈ ∞
trolA(x),wearriveatatwo-parametervariationalproblemgovernedbyopera-
tor div((ε+ k(A12 y 2))p−22)A yandatwo-parameterHammersteinequa-
− F | ∇ | ∇
tion with non-linear kernel Fε,k(y,z)=(ε+ k(y2))p−22y+(ε+ k(z2))p−22z.
F F
Finally, we deal with a two-parameter family of optimal control problems in
the coefficients for a system of elliptic boundary value problem and equation
of Hammerstein type. We consequently provide the well-posedness analysis
for the perturbed optimal control problems in Sections 5. In section 6, we
showthatthe solutionsoftwo-parametricfamily ofperturbedoptimalcontrol
4 T.Durante,O.P.Kupenko,R.Manzo
problems can be considered as appropriate approximations to optimal pairs
for the original problem similar to (1)–(4). To the end, we note that the ap-
proximation and regularization are not only considered to be useful for the
mathematicalanalysis,butalsoforthe purposeofnumericalsimulations.The
numerical analysis as well as the case of degenerating controls are subjects to
future publications.
2 Notation and preliminaries
Let Ω be a bounded open subset of RN (N 1) with a Lipschitz boundary.
≥
Let p be a real number such that 2 p < , and let q = p/(p 1) be
≤ ∞ −
the conjugate of p. Let SN := RN(N2+1) be the set of all symmetric matrices
A = [a ]N , (a = a R). We suppose that SN is endowed with the
ij i,j=1 ij ji ∈
Euclidianscalar product A B =tr(AB)=a b andwith the corresponding
ij ij
·
Euclidiannorm A SN =(A A)1/2.Wealsomakeuseoftheso-calledspectral
norm A :=sukpkAξ : ξ· RN with ξ =1 of matrices A SN, which
2
k k | | ∈ | | ∈
is different from the Euclidean norm A SN. However, the relation A 2
A SN √N A 2(cid:8)holds true for all Ak SkN. (cid:9) k k ≤
k k ≤ k k ∈
Let L1(Ω)N(N2+1) = L1 Ω;SN be the space of integrable functions whose
values are symmetric matrices. By BV(Ω;SN) we denote the space of all
matrices in L1(Ω;SN) for(cid:0)which t(cid:1)he norm
kAkBV(Ω;SN) =kAkL1(Ω;SN)+ |DA|=kAkL1(Ω;SN)
ZΩ
+ sup a divϕdx : ϕ C1(Ω;RN), ϕ(x) 1 for x Ω
ij ∈ 0 | |≤ ∈
1≤iX≤j≤N nZΩ o
(5)
is finite.
Weak Compactness Criterion in L1(Ω). Throughout the paper we will
often use the concept of weak and strong convergence in L1(Ω). Let a
{ ε}ε>0
be abounded sequenceof functions inL1(Ω). We recallthat a is called
{ ε}ε>0
equi-integrableonΩ,ifforanyδ >0thereisaτ =τ(δ)suchthat a dx<
Sk εk
δ for every measurable subset S Ω of Lebesgue measure S < τ. Then the
⊂ | | R
following assertions are equivalent for L1(Ω)-bounded sequences:
(i) a sequence a is weakly compact in L1(Ω);
{ k}k∈N
(ii) the sequence a is equi-integrable.
{ k}k∈N
Lemma 1 (Lebesgue’s Theorem) If a sequence a L1(Ω) is equi-
{ k}k∈N ⊂
integrable and a a almost everywhere in Ω then a a in L1(Ω).
k k
→ →
Lemma 2 ([28]) If a sequence ϕk k∈N is bounded in L1(Ω), ϕk 0 a.e. in
{ } →
Ω and gk k∈N is equi-integrable, then ϕk gk 0 strongly in L1(Ω).
{ } · →
OptimalcontrolsforHammersteinsystemwithanisotropicp-Laplacian 5
Lemma 3 ([28]) Let B (x,ξ) and B(x,ξ) be Caratheodory vector functions
n
acting from Ω R to R. These vector functions are assumed to satisfy the
×
monotonicity and pointwise convergence conditions
(B (x,ξ) B (x,η))(ξ η) 0, B (x,0) 0,
n n n
− − ≥ ≡
B (x,ξ) c (ξ )< ; lim B (x,ξ)=B(x,ξ),
n 0 n
| |≤ | | ∞ n→∞
for a.e. x Ω and every ξ R. If v ⇀v in Lp(Ω), B (x,v )⇀z in Lq(Ω),
n n n
∈ ∈
then
liminf B (x,v ),v z,v , (6)
n n n Lq(Ω);Lp(Ω) Lq(Ω);Lp(Ω)
n→∞ h i ≥h i
and in the case of equality in (6), we have z =B(x,v).
Admissible controls. Let ξ , ξ be given elements of L∞(Ω) BV(Ω) sat-
1 2
∩
isfying the conditions
0<α ξ (x) ξ (x) a.e. in Ω, (7)
1 2
≤ ≤
where α is a given positive value.
We define the class of admissible controls A as follows
ad
A = A L∞(Ω;SN) ξ12|η|2 ≤(η,Aη)RN ≤ξ22|η|2a.e.inΩ, ∀η ∈RN,
ad ( ∈ (cid:12) A12 BV(Ω;SN), DA21 γ, )
(cid:12) ∈ Ω| |≤
(cid:12) (8)
(cid:12) R
where γ >0 is a given con(cid:12)stant. In view of estimate
A21(x) SN √N A21(x) 2 √Nξ2(x) a.e. in Ω,
k k ≤ k k ≤
it is clear that A is a nonempty convex subset of L∞(Ω;SN).
ad
Anisotropic Laplace operator. Let us consider now the nonlinear operator
(A,y):L∞(Ω;SN) W1,p(Ω) W−1,q(Ω) defined as
A × 0 →
p−2
(A,y)= div (A y, y)RN 2 A y
A − | ∇ ∇ | ∇
or via the pairing (cid:0) (cid:1)
p−2
hA(A,y),viW−1,q(Ω);W01,p(Ω) :=ZΩ|(A∇y,∇y)RN| 2 (A∇y,∇v)RN dx
= |A12∇y|p−2(A∇y,∇v)RN dx, ∀v ∈W01,p(Ω). (9)
ZΩ
Definition 1 We say that a function y = y(A,f) is a weak solution (in the
sense of Minty) to boundary value problem
p−2
div (A y, y)RN 2 A y =f in Ω, (10)
− | ∇ ∇ | ∇
(cid:0) A∈Aad, y ∈W01,p(Ω(cid:1)), (11)
6 T.Durante,O.P.Kupenko,R.Manzo
for a fixed control A A and given function f L2(Ω) if the inequality
ad
∈ ∈
|A1/2∇ϕ|pR−N2(A∇ϕ,∇ϕ−∇y)RN dx≥ f(ϕ−y)dx (12)
ZΩ ZΩ
holds for any ϕ C∞(Ω).
∈ 0
Remark 1 Anotherdefinitionoftheweaksolutiontotheconsideredboundary
value problem appears more natural:
|A1/2∇y|pR−N2(A∇y,∇ϕ)RN dx= fϕdx∀ϕ∈C0∞(Ω).
ZΩ ZΩ
However,bothconceptsfortheweaksolutionscoincide(see,forinstance,[29]).
Let us show that for each A A operator () = (A, ) : W1,p(Ω)
∈ ad A · A · 0 →
W−1,q(Ω)isstrictlymonotone,coerciveandsemi-continuous,wheretheabove
mentioned properties have respectively the following meaning:
y v,y v 0, y,v W1,p(Ω); (13)
hA −A − iW−1,q(Ω);W01,p(Ω) ≥ ∀ ∈ 0
y v,y v =0 = y =v; (14)
hA −A − iW−1,q(Ω);W01,p(Ω) ⇒
y,y + provided y ; (15)
hA iW−1,q(Ω);W01,p(Ω) → ∞ k kW01,p(Ω)→∞
R t (y+tv),w is continuous y,v,w W1,p(Ω).
∋ 7→hA iW−1,q(Ω);W01,p(Ω) ∀ ∈ 0
(16)
Indeed, the right-hand side of (9) is continuous with respect to v W1,p(Ω)
∈ 0
and, therefore, represents an element of W−1,q(Ω) because
p−1 1
|A21∇y|p−2(A∇y,∇v)RN dx≤ |A21∇y|pdx p |A12∇v|pdx p
ZΩ (cid:18)ZΩ (cid:19) (cid:18)ZΩ (cid:19)
ξ p y p−1 v = ξ p y p−1 v
≤k 2kL∞(Ω)k∇ kLp(Ω)Nk∇ kLp(Ω)N k 2kL∞(Ω)k kW01,p(Ω)k kW01,p(Ω)
i(nwgeafrpopmlythheerectohnediHtio¨olndeAr’sineAqua)l.ityHeanncde,esftoimr aetaech|AA12∇ϕ|Ap ≤ξt2ph|∇e ϕop|percaotmor-
ad ad
(A, ) : W1,p(Ω) W−1,q∈(Ω) is bounded. The coercivi∈ty property of we
A · 0 → A
get immediately, since
y,y αp y p .
hA iW−1,q(Ω);W01,p(Ω) ≥ k kW01,p(Ω)
As for the proof ofthe strictmonotonicity and semicontinuity of the operator
,wereferforthedetailsto[22,24]).Then,bywellknownexistenceresultsfor
A
non-linearellipticequationswithcoercive,semi-continuous,strictlymonotone
operators, the Dirichlet boundary value problem (10)–(11) admits a unique
weak solution for every fixed control matrix A A and every distribution
ad
∈
f L2(Ω).
∈
On equations of Hammerstein type. Let Y and Z be Banach spaces, let
Y Y be an arbitrary bounded set, and let Z∗ be the dual space to Z. To
0
⊂
beginwithwerecallsomeusefulpropertiesofnon-linearoperators,concerning
the solvability problem for Hammerstein type equations and systems.
OptimalcontrolsforHammersteinsystemwithanisotropicp-Laplacian 7
Definition 2 We say that the operator G : D(G) Z Z∗ is radially
⊂ →
continuous if for any z ,z X there exists ε>0 such that z +τz D(G)
1 2 1 2
∈ ∈
for all τ [0,ε] and a real-valued function [0,ε] τ G(z +τz ),z
1 2 2 Z∗;Z
∈ ∋ → h i
is continuous.
Definition 3 An operator G:Y Z Z∗ is said to have a uniformly semi-
× →
bounded variation (u.s.b.v.) if for any bounded set Y Y and any elements
0
⊂
z ,z D(G) such that z R, i=1,2, the following inequality
1 2 i Z
∈ k k ≤
G(y,z ) G(y,z ),z z inf C (R; z z ) (17)
1 2 1 2 Z∗;Z y 1 2 Z
h − − i ≥−y∈Y0 k| − k|
holds true provided the function C : R R R is continuous for each
y + +
× →
1
element y Y , and C (r,t) 0 as t 0, r > 0. Here, is a
0 y Z
∈ t → → ∀ k|·k|
seminorm on Z such that is compact with respect to the norm .
Z Z
k|·k| k·k
It is worth to note that Definition 3 gives in fact a certain generalization
of the classical monotonicity property. Indeed, if C (ρ,r) 0, then (17) im-
y
≡
plies the monotonicity property for the operatorG with respect to the second
argument.
Remark 2 EachoperatorG:Y Z Z∗ withu.s.b.v.possessesthefollowing
× →
property (see for comparisonRemark 1.1.2 in [3]): if a set K Z is such that
⊂
z k and G(y,z),z k for all z K and y Y , then there
Z 1 Z∗;Z 2 0
k k ≤ h i ≤ ∈ ∈
exists a constant C >0 such that G(y,z) C, z K and y Y .
Z∗ 0
k k ≤ ∀ ∈ ∀ ∈
Let B : Z∗ Z and F : Y Z Z∗ be given operators such that the
→ × →
mapping Z∗ z∗ B(z∗) Z is linear. Let g Z be a given distribution.
∋ 7→ ∈ ∈
Then a typical operator equation of Hammerstein type can be represented as
follows
z+BF(y,z)=g. (18)
The following existence result is well-known (see [3, Theorem 1.2.1]).
Theorem 1 Let B : Z∗ Z be a linear continuous positive operator such
→
that there exists a right inverse operator B−1 :Z Z∗. Let F :Y Z Z∗
r → × →
be an operator with u.s.b.v such that F(y, ) : Z Z∗ is radially continuous
· →
for each y Y and the following inequality holds true
0
∈
F(y,z) B−1g,z 0 if only z >λ>0, λ=const.
h − r iZ∗;Z ≥ k kZ
Then the set
(y)= z Z : z+BF(y,z)=g in the sense of distributions
H { ∈ }
is non-empty and weakly compact for every fixed y Y and g Z.
0
∈ ∈
In what follows, we set Y =W1,p(Ω), Z =Lp(Ω), and Z∗ =Lq(Ω).
0
8 T.Durante,O.P.Kupenko,R.Manzo
3 Setting of the optimal control problem
Let us consider the following optimal control problem:
Minimize I(A,y,z)= z(x) z (x)2dx , (19)
d
| − |
n ZΩ o
subject to the constraints
|A1/2∇y|p−2(A∇y,∇ϕ)RN dx= fϕdx, ∀ϕ∈W01,p(Ω), (20)
ZΩ ZΩ
A A , y W1,p(Ω), (21)
∈ ad ∈ 0
zφdx+ BF(y,z)φdx=0, φ Lq(Ω), (22)
∀ ∈
ZΩ ZΩ
wheref L2(Ω) andz Lp(Ω) aregivendistributions, B :Lq(Ω) Lp(Ω)
d
is a linea∈r operator, F :W∈1,p(Ω) Lp(Ω) Lq(Ω) is a non-linear o→perator.
Letus denote by Ξ L0∞(Ω;S×N) W1→,p(Ω) Lp(Ω) the setof alladmis-
⊂ × 0 ×
sible triplets to the optimal control problem (19)–(22).
Hereinafter we suppose that the space L∞(Ω;SN) W1,p(Ω) Lp(Ω) is
× 0 ×
endowedwiththenormk(A,y,z)kL∞(Ω;SN)×W01,p(Ω)×Lp(Ω) :=kA12kBV(Ω;SN)+
y + z .
k kW01,p(Ω) k kLp(Ω)
Remark 3 Werecallthatasequence f ∞ convergesweakly-∗tof inBV(Ω)
{ k}k=1
if and only if the two following conditions hold (see [2]): f f strongly in
k
L1(Ω) and Df ⇀∗ Df weakly∗ in the space of Radon measu→res (Ω;RN).
k
M
Moreover, if f ∞ BV(Ω) converges strongly to some f in L1(Ω) and
{ k}k=1 ⊂
satisfies sup Df <+ , then (see, for instance, [2])
k∈N Ω| k| ∞
R
(i) f BV(Ω) and Df liminf Df ;
k
∈ ZΩ| |≤ k→∞ ZΩ| | (23)
∗
(ii) f ⇀f in BV(Ω).
k
Alsowerecall,thatuniformlyboundedsetsinBV-normarerelativelycompact
in L1(Ω).
Definition 4 We say that a sequence of triplets (Ak,yk,zk) k∈N from the
space L∞(Ω;SN) W1,p(Ω) Lp(Ω) τ-converges{to a triplet}(A ,y ,z ) if
× 0 × 0 0 0
A21 ⇀∗ A12 in BV(Ω;SN), y ⇀y in W1,p(Ω) and z ⇀z in Lp(Ω).
k 0 k 0 0 k 0
Further we use the following auxiliary results.
Proposition 1 For each A A and every f L2(Ω), a weak solution y to
ad
∈ ∈
variational problem (20)–(21) satisfies the estimate
q
y α−q f p . (24)
k kW01,p(Ω) ≤ k kW−1,q(Ω)
OptimalcontrolsforHammersteinsystemwithanisotropicp-Laplacian 9
Proof The estimate (24) immediately follows from the following relations
αpkykpW01,p(Ω) ≤ZΩ|A12∇y|pdx=hA(A,y),yiW−1,q(Ω);W01,p(Ω)
= f,y f y .
h iW−1,q(Ω);W01,p(Ω) ≤k kW−1,q(Ω)k kW01,p(Ω)
Lemma 4 Let (A ,y ) L∞(Ω;SN) W1,p(Ω) be a sequenceof pairs
k k ∈ × 0 k∈N
such that Ak ∈nAad ∀k ∈ N, Ak12 ⇀∗ A12 in BVo(Ω;SN), and yk ⇀ y in
W1,p(Ω). Then
0
p−2
kl→im∞ZΩ|(∇ϕ,Ak∇ϕ)RN | 2 (∇yk,Ak∇ϕ)RN dx
= |(∇ϕ,A∇ϕ)RN |p−22 (∇y,A∇ϕ)RN dx, ∀ϕ∈C0∞(Ω). (25)
ZΩ
Proof Since Ak21 →A12 in L1(Ω;SN) and {Ak}k∈N is bounded in L∞(Ω;SN),
by Lebesgue’s Theorem we get that Ak12 → A21 strongly in Lr(Ω;SN) for
every 1 ≤ r < +∞. Hence, Ak21∇ϕ → A12∇ϕ strongly in Lp(Ω)N for every
ϕ C∞(Ω). Therefore,
∈ 0
|Ak12∇ϕ|p−2Ak21∇ϕ→|A21∇ϕ|p−2A21∇ϕ in Lq(Ω)N, ∀ϕ∈C0∞(Ω). (26)
Moreover,sinceAk12∇ψ →A12∇ψstronglyinLq(Ω)N foreveryψ ∈C0∞(Ω)
and y ⇀ y in Lp(Ω)N, it follows that
k
∇ ∇
1 1
A2 y , ψ dx= y ,A2 ψ dx
ZΩ(cid:16) k∇ k ∇ (cid:17)RN ZΩ(cid:16)∇ k k∇ (cid:17)RN
→ZΩ(cid:16)∇y,A21∇ψ(cid:17)RN dx=ZΩ(cid:16)A12∇y,∇ψ(cid:17)RN dx, ∀ψ ∈C0∞(Ω) (27)
as a product of weakly and strongly convergent sequences in Lp(Ω)N and
Lq(Ω)N, respectively. Using the fact that
1
sup A2 y ξ sup y <+ ,
k∈Nk k∇ kkLp(Ω)N ≤k 2kL∞(Ω)k∈Nk∇ kkLp(Ω)N ∞
we finally get from (27)
Ak12∇yk ⇀A21∇y in Lp(Ω)N. (28)
Thus, to complete the proof it remains to note that
p−2
|(∇ϕ,Ak∇ϕ)RN | 2 (∇yk,Ak∇ϕ)RN dx
ZΩ
= A12 ϕp−2A21 ϕ,A21 y dx
ZΩ(cid:16)| k∇ | k∇ k∇ k(cid:17)RN
and apply the properties (26) and (28).
10 T.Durante,O.P.Kupenko,R.Manzo
Thefollowingresultconcernstheregularityoftheoptimalcontrolproblem
(19)–(22).
Proposition 2 Let B : Lq(Ω) Lp(Ω) and F : W1,p(Ω) Lp(Ω) Lq(Ω)
→ 0 × →
be operators satisfying all conditions of Theorem 1. Then the set
Ξ = (A,y,z) L∞(Ω;SN) W1,p(Ω) Lp(Ω):
∈ × 0 ×
(cid:8) (A,y)=f, z+BF(y,z)=0
A
is nonempty for every f L2(Ω). (cid:9)
∈
Proof Let A A be an arbitrary admissible control. Then for a given
ad
∈
f L2(Ω), the Dirichlet boundary problem (20)–(21) admits a unique so-
lut∈iony =y(A,f) W1,p(Ω)whichsatisfiesthe estimate(24).Itremainsto
A ∈ 0
remark that the corresponding Hammerstein equation z+BF(y ,z)=0 has
A
a nonempty set of solutions (y ) by Theorem 1.
A
H
4 Existence of optimal solutions
Thefollowingresultiscrucialforourconsiderationanditstatesthefact,that
thesetofadmissibletripletstotheoptimalcontrolproblem(19)–(22)isclosed
with respect to τ-topology of the space L∞(Ω;SN) W1,p(Ω) Lp(Ω).
× 0 ×
Theorem 2 Assume the following conditions hold:
– The operators B : Lq(Ω) Lp(Ω) and F : W1,p(Ω) Lp(Ω) Lq(Ω)
→ 0 × →
satisfy conditions of Theorem 1;
– The operator F(,z):W1,p(Ω) Lq(Ω) is compact in the following sense:
if y ⇀y weakly· in W10,p(Ω), t→hen F(y ,z) F(y ,z)strongly in Lq(Ω).
k 0 0 k → 0
Then for every f L2(Ω) the set Ξ is sequentially τ-closed, i.e. if a sequence
(Ak,yk,zk) Ξ∈k∈N τ-converges to a triplet (A0,y0,z0) L∞(Ω;SN)
{W1,p(Ω) Lp∈(Ω),}then A A , y = y(A ), z (y )∈, and, therefore×,
0 × 0 ∈ ad 0 0 0 ∈ H 0
(A ,y ,z ) Ξ.
0 0 0
∈
Proof Let (Ak,yk,zk) k∈N Ξ be any τ-convergent sequence of admissible
{ } ⊂
triplets to the optimal control problem (19)–(22), and let (A ,y ,z ) be its
0 0 0
τ-limit in the sense of Definition 4. We divide the rest of the proof onto two
steps.
Step 1. On this step we show that A A and y = y(A ). As follows
0 ad 0 0
∈
from Definition 4 and Remark 3, we have
A12 A12 in L1(Ω;SN), y ⇀y in W1,p(Ω), (29)
k → 0 k 0 0
1 1
A2 A2 almost everywhere in Ω, (30)
k → 0
1 1
DA2 liminf DA2 γ. (31)
ZΩ| 0|≤ k→∞ ZΩ| k|≤